From the previous definitions, the \(\alpha \) -th Riemann–Liouville fractional difference and Caputo fractional difference for \(x: \mathbb {N}_{a+1-n} \rightarrow \mathbb {R}\) , \(\alpha \in (n-1, n)\) , \(n \in \mathbb {Z}_{+}\) , \(k \in \mathbb {N}_{a+1}\)  and \(a \in \mathbb {R}\)  are defined by [1]}
\({\textstyle {}_{a}^{\mathrm {R}} \nabla _{k}^{\alpha } x(k):=\nabla ^n{}_{a}^{\rm G} \nabla _{k}^{\alpha -n} x(k), }\) 
\({\textstyle { }_{a}^{\mathrm {C}} \nabla _{k}^{\alpha } x(k):={ }_{a}^{\rm G} \nabla _{k}^{\alpha -n} \nabla ^{n} x(k) .}\) 
